Question

Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

Answer 1

Here, f(x) = cos x

At x=a, Where $aϵR$

=$\underset{h\to 0}{lim}f\left(x\right)=\underset{h\to 0}{lim}cosx=cosa$ [$\therefore$ f(x)=cosx]

f(a)=cosa

$\therefore \underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$

Thus, f(x) is continuous at x=a. But a is an arbitrary points so f(x) is continuous at all points.

So, f(x) is continuous at all points.

(ii) Here, f(x) = cosec x. Since, f(x) is not defined at $\times =n\pi ,nϵZ.$

Thus, f(x) is continuous at all points except $\times =n\pi ,nϵZ.$

(iii) Here, f(x)=sec x

Since, f(x) is not defined at $\times =(2n+1)\frac{\pi }{2},nϵZ.$

Thus, f(x) is continuous at all points except $\times =(2n+1)\frac{\pi }{2},nϵZ.$

(iv) Here, f(x) = cotx

SInce, f(x) is not defined at $\times =n\pi ,nϵZ.$

Thus, f(x) is continuous at all points excepts $\times =n\pi ,nϵZ.$